Question: Solve for $x$ and $y$ using elimination. $\begin{align*}x-y &= 3 \\ x+3y &= 2\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}-x+y &= -3\\ x+3y &= 2\end{align*}$ Add the top and bottom equations. $4y = -1$ Divide both sides by $4$ and reduce as necessary. $y = -\dfrac{1}{4}$ Substitute $-\dfrac{1}{4}$ for $y$ in the top equation. $x+ \dfrac{1}{4} = 3$ $x+\dfrac{1}{4} = 3$ $x = \dfrac{11}{4}$ The solution is $\enspace x = \dfrac{11}{4}, \enspace y = -\dfrac{1}{4}$.